Hopf surfaces in locally conformally Kähler manifolds with potential

  • Liviu OrneaEmail author
  • Misha Verbitsky
Original Paper


An LCK manifold with potential is a quotient M of a Kähler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on X by holomorphic homotheties and maps f to a function proportional to f. It is known that a compact M admits an LCK potential if and only if it can be holomorphically embedded to a Hopf manifold. We prove that any non-Vaisman, compact LCK manifold with potential contains a complex surface (possibly singular) with normalization biholomorphic to a Hopf surface H. Moreover, H can be chosen non-diagonal, hence, also not admitting a Vaisman structure.


Locally conformally Kähler Potential Hopf manifold Vaisman manifold 

2000 Mathematics Subject Classification




L.O. thanks the Laboratory for Algebraic Geometry at the Higher School of Economics in Moscow for hospitality and excellent research environment during February and April 2014, and April 2015. Both authors are indebted to Paul Gauduchon, Andrei Moroianu, and Victor Vuletescu for extremely useful disussions.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BucharestBucharestRomania
  2. 2.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  3. 3.Instituto Nacional de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrazil
  4. 4.Laboratory of Algebraic Geometry, Department of MathematicsHSE UniversityMoscowRussia

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